On scalar-type spectral operators

Spain, Philip

doi:10.1017/s0305004100046843

Cited by 8 publications

(9 citation statements)

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“…In [24] Spain has given a characterization of scalar spectral operators in terms of their C(#)-operational calculi, which may be used to determine whether a given equivalence class E of spectral measures contains a spectral measure of class X*. Seeking a generalization of Spain's result, we obtain the following sufficient condition for a class E to contain a spectral measure of class G. THEOREM …”

Section: H(f)* = I F(z)f(dz)mentioning

confidence: 94%

“…Hence, its restriction H c x to C(N) is also weakly compact. The scalar part 5 of T has the C(iV)-operational calculus H defined by = JJ(z)E(dz) (feC(N)), and so an evident modification of Spain's result [24] implies that 5 is a scalar spectral operator. If Q denotes the quasinilpotent part of T, then T = S + Q and Q commutes with 5; hence T is spectral.…”

Section: H X (F)=\ F(z)e(dz)x (Fsb(n))mentioning

confidence: 99%

“…Theorem 4 gives some necessary and sufficient conditions for operators of the latter type to be scalar prespectral of some class G. Theorem 5 presents a sufficient condition for this, whereas Example 4 shows that this condition is not necessary in general (in the case G = X*, the dual of the space X, it is, cf. Spain [24]). Example 5, which is a version of a remarkable example of Fixman [11] and of Berkson and Dowson [4], shows that there is a finitely spectral operator in l°° that is not prespectral of any class.…”

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confidence: 99%

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Finitely spectral operators

Nagy

1986

Glasgow Math. J.

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Introduction.In the theory of spectral (and prespectral) operators in a Banach space or in a locally convex topological vector space the countable additivity (in some topology) of a resolution of the identity of the operator is a standing assumption. One might wonder why. Even if one cannot completely agree with the opinion of Diestel and Uhl ([6, p. 32]) stating that "countable additivity [of a set function] is often more of a hindrance than a help", it might be interesting to study which portions of the theory of (pre)spectral operators and in which form extend to the more general situation described below.In this paper we study bounded linear operators in a complex Banach space which possess a finitely additive, uniformly bounded resolution of the identity defined on all Borel subsets of the complex plane. In the first part of the discussion (cf. Theorems 1 and 2) we summarize the most important similarities and differences between the theories of prespectral and of finitely spectral operators, and naturally omit the similar proofs. In Theorem 3 we characterize the finitely spectral operators as those having certain operational calculi. Example 3 answers a question of Gillespie [13, p. 44] in the negative: it shows that there is a nonspectral prespectral operator having a unique resolution of the identity (of any class). Then we pose the following apparently open problem: does there exist a finitely spectral operator of scalar type which is not prespectral of any class?The following parts of Section 3 may be regarded as contributions toward the solution of this problem, though they can be of interest in themselves. A finitely spectral operator of scalar type has a continuous C(/£)-operational calculus. Theorem 4 gives some necessary and sufficient conditions for operators of the latter type to be scalar prespectral of some class G. Theorem 5 presents a sufficient condition for this, whereas Example 4 shows that this condition is not necessary in general (in the case G = X*, the dual of the space X, it is, cf. Spain [24]). Example 5, which is a version of a remarkable example of Fixman [11] and of Berkson and Dowson [4], shows that there is a finitely spectral operator in l°° that is not prespectral of any class. Finally, Theorem 6 demonstrates that in a space without a copy of l°° each finitely spectral operator is spectral, whereas in a space containing a copy of /°° there are proper finitely spectral operators.

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Section: H(f)* = I F(z)f(dz)mentioning

confidence: 94%

Section: H X (F)=\ F(z)e(dz)x (Fsb(n))mentioning

confidence: 99%

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confidence: 99%

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Finitely spectral operators

Nagy

1986

Glasgow Math. J.

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“…As an improvement of this result we quote without proof the following theorem due to P. G. Spain [17].…”

Section: Definition 22 Let Kcc and A(k) Be Either A Closed Subalgebmentioning

confidence: 98%

𝑝-absolutely summing operators and the representation of operators on function spaces

Rice¹

1974

Trans. Amer. Math. Soc.

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We introduce a class of p-absolutely summing operators which we call p-extending. We show that for a logmodular function space A(K), an operator T: A(K)->Jf is p-extending if and only if there exists a probability measure ß on K such that T extends to an i some try T: AP(K, We use this result to give necessary and sufficient conditions under which a bounded linear operator is isometrically equivalent to multiplication by z on a' space L"(K, n) and certain Hardy spaces fip(Kt /x). 1. Introduction. We wish to discuss the problem of representing an operator on a Banach space as multiplication by the independent variable z on the function spaces Lp(K, li) and HP(K, p), where K is a compact subset of the complex plane and ft a positive measure on K such that li(K) = 1. Perhaps the best known instance of this arises via the spectral theorem for normal operators on a Hilbert space [7]. Given a normal operator N from a certain class called simple normal operators [4], the spectral theorem shows the existence of a positive measure p. on the spectrum (A.N) of N, with p.(a(N)) = 1, such that N is unitarily equivalent to multiplication by z on L2(erOV), p). Another important instance is provided by the subnormal operators on a Hilbert space. It is well known[l] that given any subnormal operator S having a cyclic vector, there exists a positive measure ft on the spectrum o(S) of S, with fi(a(S)) = 1 such that S is unitarily equivalent to multiplication by z on H2(cÄS), li). Such representations have proved important, for example, in the study of the invariant subspaces of these operators ([2], [18]). On a general Banach space the analogue of a normal operator is a spectral operator of scalar type (Definition 2.1), which we shall, for the sake of brevity, call a scalar operator. The restriction of such an operator to an invariant subspace will be called a subscalar operator. In §2 we introduce these operators and discuss some of their properties. In particular we show that each type has a certain

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“…It is a standard technique of spectral theory to construct an integral representation for an operator and then use this to define an extended functional calculus. For example, if X is a Banach space containing no copy of c 0 , then every map from C(σ ) to X is weakly compact; this property ensures that a map defining a C(σ ) functional calculus for an operator T ∈ L(X ) can be used to construct a spectral measure and an integral representation for T [12]. The operator T is thus scalar-type spectral and admits a B(σ (T )) functional calculus that is defined via its integral representation.…”

Section: C(σ ) Functional Calculusmentioning

confidence: 99%

Functional Calculus Extensions on Dual Spaces

Terauds¹

2009

Bull. Aust. Math. Soc.

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In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this result is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply our theorem to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one.2000 Mathematics subject classification: primary 47B40.

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On scalar-type spectral operators

Abstract: The purpose of this paper is to give two characterizations of scalar-type spectral operators.

Cited by 8 publications

References 2 publications

Finitely spectral operators

Finitely spectral operators

𝑝-absolutely summing operators and the representation of operators on function spaces

Functional Calculus Extensions on Dual Spaces

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